<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  <a href="compts_1.html#V2">compact</a> <a href="compts_1.html#V3">Hausdorff</a> <a href="waybel_9.html#NM1">TopLattice</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="waybel_0.html#NM2">net</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>







<b>assume </b><b>that </b><br/><a NAME="E1:92"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>and </b><br/><a NAME="E2:92"/><i><font color="Green">E46</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds <font color="Olive">b<sub>1</sub></font> <a href="waybel_1.html#K4">"/\"</a>  is <a href="pre_topc.html#V5">continuous</a>
 <b>and </b><br/><a NAME="E3:92"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">c<sub>2</sub></font> is <a href="waybel_0.html#V11">eventually-filtered</a>
 <b>and </b><br/><a NAME="E4:92"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="waybel_9.html#R3" target="_self">is_a_cluster_point_of</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D8">LATTICE3:def 8</a><br/>

<b>assume </b><a NAME="E5:92"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>6</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E7:92"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">c<sub>2</sub></font>
 <b>and </b><br/><a NAME="E8:92"/><i><font color="Green">E50</font></i>: 
the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>6</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="txt" href="waybel_9.html#E7:92"><i><font color="Green">E49</font></i></a></i>;<br/>
<a NAME="E10:92"/><i><font color="Green">E51</font></i>: 
 <a href="waybel_0.html#K5">downarrow</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="waybel_0.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="waybel_0.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="waybel_0.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font> </span>}</span> 
 
<b>by </b><i><a class="txt" href="waybel_9.html#E7"><i><font color="Green">Lemma7</font></i></a></i>;<br/>
<a NAME="E11:92"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="waybel_0.html#R1">is_eventually_in</a>  <a href="waybel_0.html#K5">downarrow</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="waybel_0.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="92_1"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon">c<sub>8</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E2:92_1"/><i><font color="Green">E53</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font> st <font color="Maroon">c<sub>8</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Maroon">c<sub>2</sub></font> <a href="waybel_0.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="waybel_0.html#K2">.</a> <font color="Olive">b<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="waybel_9.html#E3:92"><i><font color="Green">E47</font></i></a>, <a class="ref" href="waybel_0.html#T12">WAYBEL_0:12</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>8</sub></font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_0.html#D11">WAYBEL_0:def 11</a><br/>

<b>let </b><font color="Maroon">c<sub>9</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>

<b>assume </b><a NAME="E3:92_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>9</sub></font>
 ;<br/>

<a NAME="E4:92_1"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="waybel_0.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="waybel_0.html#K2">.</a> <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="txt" href="waybel_9.html#E2:92_1"><i><font color="Green">E53</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:92_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="waybel_0.html#K2">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K5">downarrow</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="waybel_0.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="waybel_0.html#T17">WAYBEL_0:17</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E12:92"/>
 <a href="waybel_0.html#K5">downarrow</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="waybel_0.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> is <a href="pre_topc.html#V4">closed</a>
 
<b>by </b><i><a class="txt" href="waybel_9.html#E2:92"><i><font color="Green">E46</font></i></a>, <a class="ref" href="waybel_9.html#T28" target="_self">Th28</a></i>;<br/>
<a NAME="E13:92"/><b>then </b><i><font color="Green">E53</font></i>: 
 <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K5">downarrow</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="waybel_0.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="waybel_0.html#K5">downarrow</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="waybel_0.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T52">PRE_TOPC:52</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>8</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E15:92"/><i><font color="Green">E54</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font> st <font color="Maroon">c<sub>8</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Maroon">c<sub>2</sub></font> <a href="waybel_0.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K5">downarrow</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="waybel_0.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="waybel_9.html#E11:92"><i><font color="Green">E52</font></i></a>, <a class="ref" href="waybel_0.html#D11">WAYBEL_0:def 11</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>2</sub></font> <a href="waybel_9.html#K6" target="_self">|</a> <font color="Maroon">c<sub>8</sub></font> as    <a href="yellow_6.html#M2">subnet</a> of <font color="Maroon">c<sub>2</sub></font> ;<br/>
<a NAME="E17:92"/><i><font color="Green">E55</font></i>: 
 <a href="relset_1.html#K5">rng</a> the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">c<sub>9</sub></font> <a href="tarski.html#R1">c=</a>  <a href="waybel_0.html#K5">downarrow</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="waybel_0.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="92_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>10</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:92_2"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">c<sub>9</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>11</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:92_2"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">c<sub>9</sub></font>
 <b>and </b><br/><a NAME="E4:92_2"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>11</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>9</sub></font> <b>by </b><i><a class="txt" href="waybel_9.html#E3:92_2"><i><font color="Green">E56</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>13</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E7:92_2"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font>
 <b>and </b><br/><a NAME="E8:92_2"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="ref" href="waybel_9.html#D7" target="_self">Def7</a></i>;<br/>
<a NAME="E9:92_2"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="waybel_0.html#K2">.</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K5">downarrow</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="waybel_0.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="waybel_9.html#E15:92"><i><font color="Green">E54</font></i></a>, <a class="txt" href="waybel_9.html#E8:92_2"><i><font color="Green">E59</font></i></a></i>;<br/>
<a NAME="E10:92_2"/>
<font color="Maroon">c<sub>9</sub></font> <a href="waybel_0.html#K2">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="waybel_0.html#K2">.</a> <font color="Maroon">c<sub>13</sub></font>
 
<b>by </b><i><a class="txt" href="waybel_9.html#E7:92_2"><i><font color="Green">E58</font></i></a>, <a class="ref" href="waybel_9.html#T16" target="_self">Th16</a></i>;<br/>
<b>hence </b><a NAME="E11:92_2"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K5">downarrow</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="waybel_0.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="waybel_9.html#E4:92_2"><i><font color="Green">E57</font></i></a>, <a class="txt" href="waybel_9.html#E9:92_2"><i><font color="Green">E60</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E18:92"/>
<font color="Maroon">c<sub>3</sub></font> <a href="waybel_9.html#R3" target="_self">is_a_cluster_point_of</a> <font color="Maroon">c<sub>9</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="92_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>10</sub></font> be    <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Maroon">c<sub>3</sub></font>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_9.html#D9" target="_self">WAYBEL_9:def 9</a><br/><b>let </b><font color="Maroon">c<sub>11</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>9</sub></font>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_0.html#D12">WAYBEL_0:def 12</a><br/>



<b>consider </b><font color="Maroon">c<sub>12</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E2:92_3"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font>
 <b>and </b><br/><a NAME="E3:92_3"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="ref" href="waybel_9.html#D7" target="_self">Def7</a></i>;<br/>
<a NAME="E4:92_3"/>
<font color="Maroon">c<sub>2</sub></font> <a href="waybel_0.html#R2">is_often_in</a> <font color="Maroon">c<sub>10</sub></font>
 
<b>by </b><i><a class="txt" href="waybel_9.html#E4:92"><i><font color="Green">E48</font></i></a>, <a class="ref" href="waybel_9.html#D9" target="_self">Def9</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>13</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E6:92_3"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>13</sub></font>
 <b>and </b><br/><a NAME="E7:92_3"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="waybel_0.html#K2">.</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="ref" href="waybel_0.html#D12">WAYBEL_0:def 12</a></i>;<br/>
<a NAME="E8:92_3"/>
<font color="Maroon">c<sub>8</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>13</sub></font>
 
<b>by </b><i><a class="txt" href="waybel_9.html#E3:92_3"><i><font color="Green">E57</font></i></a>, <a class="txt" href="waybel_9.html#E6:92_3"><i><font color="Green">E58</font></i></a>, <a class="ref" href="yellow_0.html#D2">YELLOW_0:def 2</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>14</sub></font> = <font color="Maroon">c<sub>13</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>9</sub></font> <b>by </b><i><a class="ref" href="waybel_9.html#D7" target="_self">Def7</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>14</sub></font>
;<br/>

<a NAME="E10:92_3"/>
<font color="Maroon">c<sub>9</sub></font> is  <a href="yellow_6.html#V2">full</a>  <a href="yellow_6.html#M1">SubNetStr</a> of <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="waybel_9.html#T14" target="_self">Th14</a></i>;<br/>
<b>hence </b><a NAME="E11:92_3"/>
<font color="Maroon">c<sub>11</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>14</sub></font>
 <b>by </b><i><a class="txt" href="waybel_9.html#E2:92_3"><i><font color="Green">E56</font></i></a>, <a class="txt" href="waybel_9.html#E6:92_3"><i><font color="Green">E58</font></i></a>, <a class="ref" href="yellow_6.html#T21">YELLOW_6:21</a></i>;<br/>

<b>thus </b><a NAME="E12:92_3"/>
<font color="Maroon">c<sub>9</sub></font> <a href="waybel_0.html#K2">.</a> <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="waybel_9.html#E7:92_3"><i><font color="Green">E59</font></i></a>, <a class="ref" href="waybel_9.html#T16" target="_self">Th16</a></i>;<br/>


</div><b>end;</b></div>
<b>then consider </b><font color="Maroon">c<sub>10</sub></font> being    <a href="yellow_6.html#M2">subnet</a> of <font color="Maroon">c<sub>9</sub></font><b> such that </b><br/><a NAME="E20:92"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="yellow_6.html#K11">Lim</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="ref" href="waybel_9.html#T32" target="_self">Th32</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>11</sub></font> being   <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>9</sub></font><b> such that </b><br/><a NAME="E22:92"/><i><font color="Green">E57</font></i>: 
the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">c<sub>10</sub></font> <a href="funct_2.html#R2">=</a> the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">c<sub>9</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>11</sub></font>
 <b>and </b><br/><a NAME="E23:92"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>9</sub></font>ex <font color="Olive">b<sub>2</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>10</sub></font> st <br/>for <font color="Olive">b<sub>3</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>10</sub></font> st <font color="Olive">b<sub>2</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">b<sub>3</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>11</sub></font> <a href="yellow_6.html#K3">.</a> <font color="Olive">b<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="yellow_6.html#D12">YELLOW_6:def 12</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>12</sub></font> =  <a href="relset_1.html#K5">rng</a> the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">c<sub>10</sub></font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<a NAME="E25:92"/>
<font color="Maroon">c<sub>12</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K5">rng</a> the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="txt" href="waybel_9.html#E22:92"><i><font color="Green">E57</font></i></a>, <a class="ref" href="relat_1.html#T45">RELAT_1:45</a></i>;<br/>
<a NAME="E26:92"/><b>then </b>
<font color="Maroon">c<sub>12</sub></font> <a href="tarski.html#R1">c=</a>  <a href="waybel_0.html#K5">downarrow</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="waybel_0.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="waybel_9.html#E17:92"><i><font color="Green">E55</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E27:92"/><b>then </b><i><font color="Green">E58</font></i>: 
 <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">c<sub>12</sub></font> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K5">downarrow</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="waybel_0.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T49">PRE_TOPC:49</a></i>;<br/>
<a NAME="E28:92"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">c<sub>12</sub></font>
 
<b>by </b><i><a class="txt" href="waybel_9.html#E20:92"><i><font color="Green">E56</font></i></a>, <a class="ref" href="waybel_9.html#T24" target="_self">Th24</a></i>;<br/>
<a NAME="E29:92"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K5">downarrow</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="waybel_0.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="waybel_9.html#E13:92"><i><font color="Green">E53</font></i></a>, <a class="txt" href="waybel_9.html#E27:92"><i><font color="Green">E58</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>13</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E31:92"/><i><font color="Green">E59</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="waybel_0.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="waybel_0.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font> )
 <b>by </b><i><a class="txt" href="waybel_9.html#E10:92"><i><font color="Green">E51</font></i></a></i>;<br/>
<a NAME="E32:92"/>
<font color="Maroon">c<sub>4</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="waybel_0.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="txt" href="waybel_9.html#E1:92"><i><font color="Green">E45</font></i></a>, <a class="txt" href="waybel_9.html#E31:92"><i><font color="Green">E59</font></i></a>, <a class="ref" href="yellow_0.html#T24">YELLOW_0:24</a></i>;<br/>
<b>hence </b><a NAME="E33:92"/>
<font color="Maroon">c<sub>4</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="waybel_9.html#E8:92"><i><font color="Green">E50</font></i></a></i>;<br/>


</div>
